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arXiv · 2210.07921

A note on the number of Egyptian fractions

Noah Lebowitz-Lockard·Victor Souza
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Abstract

Refining an estimate of Croot, Dobbs, Friedlander, Hetzel and Pappalardi, we show that for all $k \geq 2$, the number of integers $1 \leq a \leq n$ such that the equation $a/n = 1/m_1 + \dotsc + 1/m_k$ has a solution in positive integers $m_1, \dotsc, m_k$ is bounded above by $n^{1 - 1/2^{k-2} + o(1)}$ as $n$ goes to infinity. The proof is elementary.

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Noah Lebowitz-Lockard, Victor Souza. 2022-10-11. A note on the number of Egyptian fractions. https://arxiv.org/abs/2210.07921

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